Rigid-analytic geometry and the uniformization of abelian
varieties
Mihran Papikian
Abstract. The purpose of these notes is to introduce some basic notions
of rigid-analytic geometry, with the aim of discussing the non-archimedean
uniformizations of certain abelian varieties.
1. Introduction
The methods of complex-analytic geometry provide a powerful tool in the study
of algebraic geometry over C, especially with the help of Serre’s GAGA theorems
[S1]. For many arithmetic questions one would like to have a similar theory over
other fields which are complete for an absolute value, namely over non-archimedean
fields. If one tries to use the metric of a non-archimedean field directly, then such
a theory immediately runs into problems because the field is totally disconnected.
It was John Tate who, in early 60’s, understood how to build the theory in
such a way that one obtains a reasonably well-behaved coherent sheaf theory and
its cohomology [T1]. This theory of so-called rigid-analytic spaces has many results which are strikingly similar to algebraic and complex-analytic geometry. For
example, one has the GAGA theorems over any non-archimedean ground field.
Over the last few decades, rigid-analytic geometry has developed into a fundamental tool in modern number theory. Among its impressive and diverse applications one could mention the following: the Langlands correspondence relating automorphic and Galois representations [D], [HT], [LRS], [Ca]; Mumford’s
work on degenerations and its generalizations [M2], [M3], [CF]; Ribet’s proof that
Shimura-Taniyama conjecture implies Fermat’s Last Theorem [Ri]; fundamental
groups of curves in positive characteristic [Ra]; the construction of abelian varieties with a given endomorphism algebra [OvdP]; and many others.
The purpose of these notes is to informally introduce the most basic notions
of rigid-analytic geometry, and, as an illustration of these ideas, to explain the
analytic uniformization of abelian varieties with split purely toric reduction. The
construction of such uniformizations is due to Tate and Mumford. In our exposition
we follow [Co1] and [FvdP].
Acknowledgements. I thank the organizers for their invitation to participate
in the conference, and for creating stimulating atmosphere during the conference.
2000 Mathematics Subject Classification. 14G22, 11G10, 11G07.
1
2
MIHRAN PAPIKIAN
I also thank Brian Conrad and the anonymous referee for useful comments on the
earlier versions of this article.
2. Tate curve
As a motivation, and to have a concrete example in our later discussions, we
start with elliptic curves. Here, the non-archimedean uniformization of so-called
Tate curves can be constructed very explicitly by using Weierstrass equations. (Of
course, such an equation-based approach does not generalize to higher dimensional
abelian varieties.) Later on we will return to the example of Tate curves, and will
see how Tate’s theorem can be recovered from a more sophisticated point of view.
Let K be a field complete with respect to a non-trivial non-archimedean valuation which we denote by | · |. Let R = {z ∈ K | |z| ≤ 1} be the ring of integers in
K, m = {z ∈ K | |z| < 1} be the maximal ideal, and k = R/m be the residue field.
For example, we can take K = Qp , R = Zp , m = pZp and k = Fp , or K = Fq ((t)),
R = Fq [[t]], m = tFq [[t]] and k = Fq . The field K need not necessarily be locally
compact or discretely valued, for example, it can be C((t)) or the completion of an
algebraic closure of Qp . Nevertheless, if it simplifies the exposition, we shall sometimes assume that the underlying field is either algebraically closed or discretely
valued.
Example 2.1. Recall that explicitly Qp is the field of “Laurent series”
(∞
)
X
i
ai p | n ∈ Z, ai ∈ Z, 0 ≤ ai ≤ p − 1 .
i=n
The norm of α =
if α 6= 0)
P∞
i=n
ai pi is given by (without loss of generality we assume an 6= 0
(
p−n
|α| =
0
if α 6= 0,
otherwise.
The name “non-archimedean” comes from the fact that for such fields the usual
triangle inequality holds in a stronger form: |α + β| ≤ max(|α|, |β|).
Let E be an elliptic curve over C. Classically it is known that
(2.1)
E(C) ∼
= C/Λ ∼
= C/(Z ⊕ Zτ ),
where Im(τ ) > 0, with the isomorphism given in terms of the Weierstrass ℘-function
and its derivative; see [Si1, Ch.V]. To motivate the formulae over K, let
u = e2πiz ,
q = e2πiτ ,
q Z = {q m | m ∈ Z},
and consider the complex-analytic isomorphism
(2.2)
∼
C/Λ −→ C× /q Z
z 7−→ u.
Note that since Im(τ ) > 0, we have |q| < 1. We use q-expansion to explicitly
describe the isomorphism E(C) ∼
= C× /q Z resulting from (2.1) and (2.2).
RIGID-ANALYTIC GEOMETRY
3
Theorem 2.2. Define the quantities
X nk q n
sk (q) =
∈ Z[[q]],
1 − qn
n≥1
a4 (q) = −5s3 (q),
X(u, q) =
X
n∈Z
Y (u, q) =
X
n∈Z
2
a6 (q) = −
5s3 (q) + 7s5 (q)
,
12
qn u
− 2s1 (q),
(1 − q n u)2
(q n u)2
+ s1 (q),
(1 − q n u)3
Eq : y + xy = x3 + a4 (q)x + a6 (q).
Then Eq is an elliptic curve, and X and Y define a complex-analytic isomorphism
(2.3)
C× /q Z −→ Eq (C)
(
(X(u, q), Y (u, q))
u 7−→
O
Proof. See [Si2, Ch.V].
if u 6∈ q Z ,
if u ∈ q Z .
¤
If we replace C by K and try to parametrize an elliptic curve E over K by
a group of the form K/Λ, then we run into a serious problem: K need not have
non-trivial discrete subgroups. For example, if Λ ⊂ Qp is any non-zero subgroup
and 0 6= λ ∈ Λ, then pn λ ∈ Λ for all n ≥ 0, so 0 is an accumulation point of Λ
(in p-adic topology!). If the characteristic of K is positive then “lattices” do exist.
For example, Λ = Fq [t] is an infinite discrete subgroup of the completion C∞ of an
algebraic closure of Fq (( 1t )). However, the quotient C∞ /Λ is not an abelian variety;
in fact, one can show C∞ /Λ ∼
= C∞ . (Such quotients are very interesting since they
naturally lead to the theory of Drinfeld modules [D], but this is not what we want.)
Tate’s observation was that the situation can be salvaged if one uses the multiplicative version of the uniformization. Indeed, K × has lots of discrete subgroups,
as any q ∈ K × with |q| < 1 defines the discrete subgroup q Z = {q m | m ∈ Z} ⊂ K × .
Theorem 2.3 (Tate). Let q ∈ K × with |q| < 1. Let a4 (q), a6 (q) be defined as
in Theorem 2.2. (This makes sense since a4 (q), a6 (q) ∈ Z[[q]].) The
Pseries defining
a4 and a6 converge in K. (Thanks to non-archimedean topology, n≥0 αn is convergent if |αn | → 0.) Hence, we can define Eq over K in terms of the Weierstrass
equation as in Theorem 2.2. The map (2.3) defines a surjective homomorphism
φ : L× → Eq (L)
with kernel q Z , where L is any algebraic extension of K. Moreover, when L is
Galois over K, this homomorphism is compatible with respect to the action of the
Galois group Gal(L/K).
Proof. See [T2].
¤
Gal(Qalg
p /Qp ),
Remark 2.4. The absolute Galois groups of local fields, e.g.
play a prominent role in arithmetic geometry. The Galois equivariance of nonarchimedean uniformizations, as in Tate’s theorem, makes such uniformizations
quite useful in the study of these Galois groups.
4
MIHRAN PAPIKIAN
Over the complex numbers we know that every elliptic curve E/C is isomorphic
to Eq for some q ∈ C× . In the non-archimedean case, using q-expansions, we see
that a4 (q), a6 (q) ∈ m because q ∈ m. Hence, the reduction E q of Eq over the
residue field is given by x2 + xy = x3 . In particular, E q has a node and the slopes
of the tangent lines at the node are in k. Elliptic curves with this property are said
to have split multiplicative reduction over K. We conclude that not every elliptic
curve over K can be isomorphic to some Eq – a necessary condition is that E must
have split multiplicative reduction. This also turns out to be a sufficient condition,
i.e., if E has split multiplicative reduction then there is a unique q ∈ K × with
|q| < 1 such that Eq ∼
= E. This last statement again can be proven by using the
explicit nature of the uniformization: First, E has multiplicative reduction over K
if and only if |j(E)| > 1. Next, one writes down the j-invariant of Eq as an infinite
Laurent series in q with integer coefficients j = q −1 + 744 + 196884 · q + · · · , and,
by carefully analyzing these series, shows that j(q) defines a bijection between the
two sets {q ∈ K × | |q| < 1} and {z ∈ K | |z| > 1}. Finally, one shows that E
with j(E) = α, |α| > 1, is isomorphic over K to Eq with j(q) = α if and only if E
has split multiplicative reduction. To prove a similar statement for general abelian
varieties one needs to use more sophisticated arguments. We will return to this in
§4.5.
We would like to understand the higher-dimensional generalizations of Theorem
2.3; we also would like to have an honest “analytic object” E an with underlying
set equal to the set of closed points of E and an isomorphism of “analytic spaces”
Z
E an ∼
= Gan
m,K /q in an appropriate geometric category. To do all of this we need
rigid-analytic spaces – the subject of the next section.
3. Rigid-analytic geometry
As in §2, let K be a non-archimedean field. We would like to define analytic
geometry over K. The attempt to construct such a theory in a straightforward
manner, by using the topology on K, does not work since K is totally disconnected.
To see why this is so, let a ∈ K and let D(a, r) = {x ∈ K| |x − a| ≤ r} be
the closed disc around a of radius r, where r ∈ R is a value of | · | on K × . Let
D(a, r− ) = {x ∈ K| |x−a| < r} be the open disc, and C(a, t) = {x ∈ K| |x−a| = t}
be the circle of radius t around a. Then a counterintuitive thing happens, namely
C(a, t) is an open subset of K! Indeed, for y ∈ C(a, t), the non-archimedean
S
nature of the norm implies D(y, t− ) ⊂ C(a, t). Since D(a, r) = t≤r C(a, t), the
disc D(a, r) is both open and closed. (It is worth mentioning that for some other
questions, such as the theory of Lie groups over non-archimedean fields, the total
disconnectedness of K does not cause serious problems; see [S2] where the theory
of Lie groups is developed uniformly for any ground field K, complete with respect
to a non-trivial absolute value.)
Tate’s idea for constructing a theory of analytic spaces over K was to imitate
the construction of algebraic varieties: an algebraic variety over a field K is obtained
by gluing affine varieties over K with respect to Zariski topology. Rigid-analytic
spaces over K are formed in a similar way, by gluing “affinoids” with respect to a
certain Grothendieck topology.
We start by introducing the Tate algebra Tn (K), which plays a role similar to
the role of finitely generated polynomial algebra K[Z1 , · · · , Zn ] in algebraic geometry. Then we define the notion of an affinoid, which is the analogue of a finitely
RIGID-ANALYTIC GEOMETRY
5
generated algebra over K. Finally, we explain how to “glue” the affinoids to construct rigid-analytic spaces. The standard references for this section are [T1],
[BGR] and [FvdP].
3.1. Tate algebra. The Tate algebra Tn (K) over K is the algebra of formal power series in Z1 , Z2 , . . . , Zn with coefficients in K satisfying the following
condition:
Tn (K) = KhZ1 , . . . , Zn i


X
=
as Z1s1 · · · Znsn

n
s=(s1 ,...,sn )∈N
¯
¯
¯
¯


lim
s1 +···+sn →∞
|as | = 0

.
One can think of Tn as the algebra of holomorphic functions on the polydisc
Dn (K) := {(z1 , . . . , zn ) ∈ K n | |zi | ≤ 1 for i = 1, . . . , n}.
The norm on K can be extended to a norm on Tn by:
°
°
°X
°
°
s1
sn °
(3.1)
as Z1 · · · Zn ° = maxn |as |.
°
° n
° s∈N
s∈N
Proposition 3.1. Tn (K) is a Banach algebra with respect to k·k. That is, for
any f, g ∈ Tn and any a ∈ K we have
(1) kf k ≥ 0 with equality if and only if f = 0,
(2) kf + gk ≤ max (kf k , kgk),
(3) ka · f k = |a| · kf k,
(4) k1k = 1,
(5) kf · gk ≤ kf k · kgk,
(6) Tn is complete with respect to k·k.
It is easy to see that the norm on Tn is in fact multiplicative: kf · gk = kf k·kgk.
Also note that Tn is the completion of the polynomial algebra K[Z1 , . . . , Zn ] with
respect to this norm. Some of the statements in the next proposition can be proven
by using an analogue of the Weierstrass preparation theorem, cf. Theorem 3.2.1 in
[FvdP].
Proposition 3.2.
(1) Tn is a regular noetherian unique factorization domain whose maximal
ideals all have height n.
(2) If I ⊂ Tn is a proper ideal then there exists a finite injective morphism
Td → Tn /I with d the Krull dimension of Tn /I.
(3) For any maximal ideal M ⊂ Tn , the field Tn /M is a finite extension of K
and the valuation on K uniquely extends to Tn /M .
(4) Every ideal in Tn is closed with respect to k·k.
(5) Tn is a Jacobson ring, i.e., the radical of an ideal I is equal to the intersection of all the maximal ideals containing I.
3.2. Affinoid algebras. Let I ⊂ Tn be an ideal. The algebra A = Tn /I
is called a K-affinoid algebra. Since I is closed in Tn , the algebra A is noetherian
and is a Banach algebra with respect to the quotient norm on A, i.e.,
° °
°f¯° = inf{kf + gk | g ∈ I}. It can be shown that the K-Banach topology on A
6
MIHRAN PAPIKIAN
is unique and that all K-algebra maps between K-affinoid algebras are continuous.
For simplicity of the exposition we will always assume that A is reduced.
Let X = Z(I) ⊂ Max(Tn ) be the zero set of I. By restriction, the elements
of Tn can be regarded as functions on X. Since Tn is Jacobson and A is assumed
to be reduced, the element f ∈ Tn is 0 on X if and only if f ∈ I. In particular,
the elements of A can then be regarded as (holomorphic) functions on X. The
canonical surjection Tn → A induces a map between the sets of maximal ideals
Max(A) → Max(Tn ). This map identifies X and Max(A), so the elements of A can
be regarded as functions on Max(A). As with algebraic varieties over an abstract
field, the value f (x) of f ∈ A at x ∈ X is the image of f in the residue field A/mx
at x, where mx denotes the maximal ideal of A corresponding to x.
Definition 3.3. A subset U ⊆ X is an affinoid subset of X if there is a
homomorphism of K-affinoid algebras ϕ : A → B with U = ϕ∗ (Max(B)), and
moreover, for every homomorphism ψ : A → C such that ψ ∗ (Max(C)) ⊆ U there is
a unique homomorphism of K-affinoid algebras φ : B → C with ψ = φ ◦ ϕ. (Note
that the analogous condition on finite type maps between noetherian affine schemes
corresponds to open immersions.)
It is not hard to show that we can provide the set X = Max(A) with a
Grothendieck topology, where the admissible opens are the affinoid subsets and
the admissible coverings are the coverings by affinoid subsets which have finite subcoverings. One essentially needs to check that: (1) The intersection of two affinoid
subsets in X is again an affinoid subset of X; (2) If U ⊆ X is an affinoid subset of
X and W ⊆ U is an affinoid subset of U then W is an affinoid subset of X. For
the convenience of the reader, we recall the definition of Grothendieck topology;
see [BGR, §9.1] for more details.
Definition 3.4. Let X be a set. To endow X with a Grothendieck topology
means to specify a family G of subsets of X, and for each U ∈ G a set Cov(U ) of
coverings of U by elements of G, satisfying the following conditions:
(1) ∅, X ∈ G;
(2) If U, V ∈ G then U ∩ V ∈ G;
(3) {U } ∈ Cov(U );
(4) If U, V ∈ G with V ⊂ U , and U ∈ Cov(U ), then U ∩ VS ∈ Cov(V );
(5) If U ∈ G, {Ui }i∈I ∈ Cov(US
) and Ui ∈ Cov(Ui ), then i∈I Ui ∈ Cov(U );
(6) If {Ui }i∈I ∈ Cov(U ) then i∈I Ui = U .
The elements of G are called admissible opens, and the elements of Cov(U ) are
called admissible coverings. One defines presheaves (sheaves, Čech cohomology) for
a Grothendieck topological space in a usual manner.
Given an admissible open U ⊆ X with U = Max(B), let OX (U ) = B. The map
U 7→ OX (U ) from admissible opens of X to affinoid K-algebras defines a presheaf
O on X. It is a consequence of rather non-trivial results of Gerritzen, Grauert and
Tate that O is in fact a sheaf.
Definition 3.5. The Grothendieck topological space Max(A) with the structure sheaf O is called a K-affinoid space and is denoted Sp(A).
Remark 3.6. Strictly speaking, for the purposes of the next definition we
should have defined a more general notion of “admissible opens” in an affinoid space,
RIGID-ANALYTIC GEOMETRY
7
called rational subsets. The structure sheaf canonically extends to this “stronger”
topology, and in what follows it is understood that admissibles in affinoid spaces
are to be taken in this more general sense. Nevertheless, for practical purposes it
suffices to think with affinoid subsets, and in fact affinoid subsets of Sp(A) serve as
a “base for the topology”, quite similar to open balls in a manifold.
By gluing the affinoids along admissible opens, we obtain more general spaces:
Definition 3.7. A rigid-analytic space is a pair (X, O), where X is a Grothendieck topological space and O is a sheaf of K-algebras on X, such that there exists
an admissible covering {Xi } of X with (Xi , O|Xi ) being an affinoid space.
In all examples in this section we assume that K is algebraically closed (this is
done for expository simplicity and for no other reason). The same examples can be
pushed through for general K once one has a suitable language for working with
rigid spaces on which there are non-rational points.
Example 3.8. For any affinoid space Sp(A), the set Max(A) can be identified
with the set of K-algebra homomorphisms A → K. For example, the underlying
topological space of Sp(Tn ) is the polydisc Dn (K). Indeed, any K-homomorphism
KhZ1 , . . . , Zn i → K is uniquely determined by the images of Zi ’s, which we have
to (and can) send to some point in the polydisc (note that for z ∈ K n the sums
f (z) are convergent for all f ∈ Tn exactly when z ∈ Dn (K)).
Example 3.9. Let a, b ∈ K satisfy 0 < |a| ≤ |b| ≤ 1. It is not hard to
check that the set {z ∈ K | |a| ≤ |z| ≤ |b|} is an affinoid subset of B = Sp KhZi
corresponding to the affinoid algebra KhZ, U, W i/(ZW − a, Z − bU ). The covering
B = U1 ∪ U2 , where U1 = {z ∈ K | |z| ≤ |a|} and U2 = {z ∈ K | |a| ≤ |z| ≤ 1}, is
an admissible covering.
To emphasize the importance of the requirement that admissible coverings admit finite subcoverings by affinoid subsets, consider the following covering of B. Let
r1 < r2 < · · · < 1 be real numbers in the image of the norm on K such that ri
converge
S to 1. Let Uj = {z ∈ K | |z| ≤ rj }, and let ∂B = {z ∈ K | |z| = 1}, so
B = ( j Uj ) ∪ ∂B. If we had allowed such a covering, then B would be disconnected
(something we do not want). Fortunately, since this covering obviously has no finite
subcovering, it is not an admissible covering of the affinoid space B. The notion
of admissible coverings is introduced to rule out such unpleasant phenomena as
the unit disc being disconnected, and in fact if A is a K-affinoid algebra with no
idempotents then Sp(A) is connected (in the sense that for any admissible covering
{U1 , U2 }, with U1 and U2 non-empty, the intersection U1 ∩ U2 is non-empty).
Remark 3.10. As an instructive analogue of the idea behind the definition of
a K-affinoid space, the reader is invited to try to make the totally disconnected
space {q ∈ Q | 0 ≤ q ≤ 1} acquire the compactness and connectedness properties
of [0, 1] by restricting the class of “opens” and “covers”.
Example 3.11. We can “analytify” algebraic K-schemes, much as is done for
algebraic varieties over C. Let X = Spec K[Z1 , . . . , Zn ]/(f1 , . . . , fs ) be a reduced
affine scheme of finite type over K. The topological space X can be identified with
a closed (in Zariski topology) subset of K n :
X = {z = (z1 , . . . , zn ) ∈ K n | 0 = f1 (z) = · · · = fs (z)}.
8
MIHRAN PAPIKIAN
Choose π ∈ K × with |π| < 1. For m ≥ 1 let
Xm = {z ∈ X | |zi | ≤ |π|−m for 1 ≤ i ≤ n},
so Xm has a natural structure of an affinoid (that is in fact reduced):
Xm = Sp Khπ m Z1 , . . . , π m Zn i/(f1 , . . . , fs ).
The affinoids Xm can be glued together in an obvious manner, where Xm is an
admissible open in Xm+1 . One can check that this gives a well-defined rigid-analytic
space X an .
As a more concrete example, let’s consider (A1K )an . Pick c ∈ K × with |c| > 1.
Let Br = Sp Kh cZr i := {z ∈ K| |z| ≤ |c|r } for r = 0, 1, 2, . . . Then
B0 ,→ B1 ,→ · · · Br ,→ Br+1 ,→ · · ·
zr /c ← zr+1
are glued together as indicated to give A1,an
K .
Now given a general separated (reduced) scheme X over K, we can associate
to it a (reduced) rigid-analytic space X an by considering an open affine cover of
X. Since the intersection of two open affines in X is again an open affine, we can
analytify each open affine and glue them along the intersections.
Of course, one can show that X 7→ X an is a functor from the category of
separated schemes of finite type over K into the category of rigid-analytic spaces.
The space X an represents the functor which to a rigid-analytic space X associates
the set of morphisms HomK (X , X) of Grothendieck topological spaces over K.
That this functor is representable can be proven along the lines of Theorem 1.1 in
SGA 1, Exposé XII. One eventually reduces the question to the representability
of X 7→ HomK (X , A1K ). This last functor is representable by a rigid space with
a distinguished global function, and the space A1,an
K , which we constructed above,
via gluing of balls,
has this universal property. In view of the construction of A1,an
K
to prove its universal property one has to show that on any affinoid space every
global function is bounded. Such boundedness is a fundamental property of affinoid
algebras which is not hard to verify; cf. [FvdP, §3.3]. Most of the properties of
morphisms and spaces with respect to analytification over C, as proved in the above
mentioned Exposé, carry over to the rigid-analytic setting. For example, X 7→ X an
is fully faithful on proper schemes; cf. [FvdP, Ch.4].
Example 3.12. Now let’s return to the Tate curve example and consider it
from the rigid-analytic point of view. Let Gm,K = Spec K[Z, W ]/(ZW − 1) be the
×
algebraic torus, and let Gan
with |q| < 1. We
m,K be its analytification. Let q ∈ K
an
can identify q with the automorphism of the rigid space Gm,K , given by z 7→ zq.
The action of the group Γ = q Z of the rigid space Gan
m,K is discontinuous. This
means that Gan
has
an
admissible
covering
{X
}
such
that for each i the set
i
m,K
{γ ∈ Γ | γXi ∩ Xi 6= ∅} is finite. An example of such a covering is {Xi }i∈Z , where
the affinoid subset Xn of Gan
m,K is the annulus
Xn = {z ∈ K × | |q|(n+1)/2 ≤ |z| ≤ |q|n/2 }.
Note that Xn ∩ Xn−1 = {z ∈ K × | |z| = |q|n/2 } is also an affinoid subset, and q acts
on this covering by Xn 7→ Xn+2 . Now we can form the quotient E = Gan
m,K /Γ in the
usual manner, i.e., by gluing annuli appropriately. The Grothendieck topology on E
is defined as follows: U ⊂ E is admissible if there is an admissible open V ⊂ Gan
m,K
RIGID-ANALYTIC GEOMETRY
9
such that under the canonical projection Gan
m,K → E the set V maps bijectively to
U . For example, X0 and X1 map bijectively into E and form a covering
U0 ∪ U1 of
S
this space. The intersection U0 ∩ U1 is the image of (X−1 ∩ X0 ) (X0 ∩ X1 ). The
structure sheaf O on E is defined by O(∅) = {0}, O(E) = K, O(Ui ) = O(Xi ) and
O(U0 ∩ U1 ) = O(X−1 ∩ X0 ) ⊕ O(X0 ∩ X1 ).
One checks that this indeed gives a well-defined structure of a rigid-analytic space
on E. Moreover, it is possible to show that E ∼
= Eqan . This is done by carefully
analyzing the field of meromorphic functions M(E) on E and showing M(E) =
K(X(z, q), Y (z, q)). For the details we refer to [FvdP, §5.1].
4. Uniformization of abelian varieties
In this section we show that the classical theory of analytic uniformization
of abelian varieties over C has a good analogue in the category of rigid-analytic
g
spaces. We study the quotients of (Gan
m,K ) by discrete rank-g lattices, and we
consider questions of algebraicity of such quotients. As a motivation, we first recall
the well-known corresponding theorem over C.
∼ Cg be a finite-dimensional complex vector space
4.1. Complex case. Let V =
over C. A map H : V ×V → C is a Hermitian form if it is linear in the first variable,
anti-linear in the second variable, and H(u, v) = H(v, u). The Hermitian form H
is positive-definite if H(u, u) > 0 for all u ∈ V \ {0}. By a lattice Λ in V we
mean a discrete subgroup isomorphic to Z2g . A Riemann form on G = V /Λ is a
positive-definite Hermitian form on V such that Im(H) is Z-valued on Λ × Λ.
Theorem 4.1. The quotient G = V /Λ is the analytification of a complex
abelian variety if and only if G possesses a Riemann form.
G is an abelian variety if and only if it can be embedded into some projective
space PN
C . (The image of G is an algebraic group variety according to Serre’s
GAGA.) This translates into a question of existence of line bundles on Cg with
certain properties, which then can be shown to be equivalent to the existence of
a Riemann form. Relatively few G have such a form, hence not all of them are
algebraic. For the details we refer to [M1, Ch.1].
4.2. Non-archimedean case. As with Tate curves, we consider the multiplicative version of the construction in §4.1.
Let T = Ggm,K = Spec K[Z1 , Z1−1 , . . . , Zg , Zg−1 ] be the algebraic g-dimensional
torus. Let T an be the rigid-analytic space corresponding to T . There is a natural
group homomorphism ` : T an → Rg given by `(z) = (− log |z1 |, . . . , − log |zg |). A
lattice Λ in T an is a torsion-free, Z-rank g subgroup of T (K) which is discrete in
T an , i.e., the intersection of each affinoid with Λ is finite. Equivalently, `(Λ) is a
rank-g lattice in Rg . We would like to give G = T an /Λ a structure of a rigid-analytic
space.
For simplicity assume the valuation on K is discrete, and choose a basis for
T an and Rg such that `(Λ) = Zg . Consider the unit cube
S := {(x1 , . . . , xg ) ∈ Rg | |xi | ≤ 1/2 for all i},
so Rg is covered by {a + S} with a ∈ Zg . The set U := `−1 (S) is an affinoid
`−1 (`(λ) + S), are
subspace of T an . The translates of U by elements of Λ, λU := S
an
again affinoid subsets and give an admissible covering T = λ∈Λ λU . We use
10
MIHRAN PAPIKIAN
the covering of G by the images Vλ := pr(λU ) to endow it with a structure of a
rigid-analytic space. There is a notion of proper rigid-analytic space, and it is not
hard to show that the group space G is proper. The Riemann form condition in
rigid setting translates into the following:
Theorem 4.2. G is an abelian variety if and only if there is a homomorphism
H : Λ → Λ∨ := Hom(T an , Gan
m,K )
such that H(λ)(λ0 ) = H(λ0 )(λ), and the symmetric bilinear form on Λ × Λ
Λ×Λ→Z
λ, λ0 7→ hλ, λ0 i := − log |H(λ0 )(λ)|
is positive definite.
The strategy of the proof of this theorem is very similar to the one over C.
First, we have the following analogue of the Appell-Humbert theorem.
Proposition 4.3. There is a functorial isomorphism of groups
Pic(G) ∼
= H1 (Λ, O(T an )× ),
α
where O(T an )× = {β · Z1α1 · · · Zg g | β ∈ K × and (α1 , . . . , αg ) ∈ Zg } is the multiplicative group of nowhere-vanishing holomorphic functions on T an , and Λ ⊂
T an (K) acts through its translation action on T an . Moreover, every element in
H1 (Λ, O(T an )× ) can be uniquely represented by a 1-cocycle Zλ (z) = d(λ)H(λ)(z),
where
H : Λ → Λ∨ = {Z1α1 · · · Zgαg | (α1 , . . . , αg ) ∈ Zg }
is a group homomorphism and d : Λ → K × is a map satisfying
d(λ1 λ2 )d(λ1 )−1 d(λ2 )−1 = H(λ2 )(λ1 ).
Proof. The proof is essentially the same as over C [M1, Ch.1], using the fact
[FvdP, Ch.VI] that the line bundles on T an are trivial.
¤
Thus, every line bundle L on G corresponds to a cocycle Zλ and every such
cocycle is given by a pair (H, d). That is, every line bundle corresponds to a pair
(H, d), and we will denote this line bundle by L(H, d). One easily checks that
αg
1
L(H1 , d1 ) ∼
= L(H2 , d2 ) if and only if H1 = H2 and d1 (λ)/d2 (λ) = λα
1 . . . λg for
g
some fixed (α1 , . . . , αg ) ∈ Z .
The global sections of a line bundle L on G can be explicitly described using
theta series and their Fourier expansions (this relies on the explicit description
of L given above). It turns out that L(H, d) is ample if and only if − log |H| is
positive definite. The positive definiteness is used to show that enough explicit
formal Fourier series are convergent, and hence give honest sections, so that L⊗3
is very ample. Finally, one uses rigid-analytic GAGA theorems, proved by Kiehl,
to conclude that G is algebraic (every closed analytic subspace of Pn,an
is the
K
analytification of a unique closed subscheme of PnK ).
Given an abelian variety A of dimension g over K it is natural to ask whether
there is an isomorphism Aan ∼
= T an /Λ for some lattice Λ. If there is such an
isomorphism then we say that A admits analytic uniformization. The corresponding
question over C has an affirmative answer, i.e., all abelian varieties over C have an
RIGID-ANALYTIC GEOMETRY
11
analytic uniformization. Indeed, let VA = H0 (A, Ω1 )∨ be the tangent space to A at
the identity, and let ∆A = H1 (A, Z). There is a natural injective homomorphism
∆A → VA
Z
γ 7→ (ω 7→
ω),
γ
where ω ∈ H0 (A, Ω1 ). The quotient VA /∆A , which is isomorphic to (C× )g /Λ via
exp, can be identified with A(C); see [M1, Ch.1]. We already saw on the example of the Tate curve that the same question over non-archimedean fields has a
more complicated answer. To characterize the abelian varieties which admit analytic uniformization one needs three fundamental concepts: Néron models, formal
schemes with their “generic fibre” functor, and the notion of analytic reduction of
a rigid-analytic space. We will only outline the ideas involved.
4.3. Analytic reductions. Given a rigid-analytic space X and an admissible
affinoid covering U having certain good properties, one can associate to it a locally
finite type scheme X over k, called the analytic reduction of X with respect to U .
First, let X = Sp(A). Denote by A◦ the R-algebra {f ∈ A | kf k ≤ 1}. There
is a natural ideal A◦◦ = {f ∈ A | kf k < 1}. The canonical analytic reduction of
X is the k-scheme X = Spec(A), where A = A◦ /A◦◦ . This can be proved to be a
reduced scheme of finite type over the residue field. Moreover, there is a canonical
surjective map of sets Max(A) → Max(A). This last map is constructed as follows
(for simplicity assume K is algebraically closed). Using an analogue of the maximum
modulus principle, it is possible to show that for f ∈ A, kf k = max|f (x)|. Hence
x∈X
under the quotient map A → A/mx = K given by evaluation at x ∈ X, A◦ maps to
R and A◦◦ maps to m. This induces a k-algebra homomorphism A → k. Since any
such homomorphism uniquely corresponds to some x ∈ Max(A), we get the map
Max(A) → Max(A), x 7→ x.
S
Next, if X is an analytic space and X = Ui is an admissible covering such
that the maps Ui ∩ Uj ⇒ U i , U j are open immersions for all i and j, then the
canonical reductions U i can be glued together along Ui ∩ Uj ’s to give a k-scheme
S
X (depending on the affinoid covering Ui ) with a surjective map of sets X → X.
In the examples of this subsection we assume that K is algebraically closed.
As in §3.2, this is done for expository purposes only, and all conclusions in the
examples are valid for general K.
Example 4.4. Let A = T1 = KhZi be the 1-dimensional Tate algebra. From
the definition of the norm (3.1) on Tn it is clear that A◦ = RhZi and A◦◦ = mhZi.
Since any power series in A◦ , up to a finite number of summands, belongs to A◦◦ , we
get A = k[Z]. The topological space of Sp(A) is the unit disc D = {z ∈ K | |z| ≤ 1}.
It is easy to see that the evaluation at z of the elements of A induces the evaluation
at z = z (mod m) of the elements of k[Z]. Hence under Max(A) → Max(A) the
interior D0 = {z ∈ K | |z| < 1} of D maps to the origin 0 of A1k , and the boundary
∂D = {z ∈ K | |z| = 1} maps onto A1k − {0}. A similar argument shows that
T n = k[Z1 , . . . , Zn ].
Example 4.5.½ Let A = KhZ, U i/(ZU − 1)¾= KhZ, Z −1 i. As a power series
°P
°
P
n°
n
°
ring A is given by
=
n∈Z an Z
n∈Z an Z | lim |an | = 0 , with the norm
n→±∞
12
MIHRAN PAPIKIAN
max |an |. The underlying topological space is {z ∈ K | |z| = 1}. As in Example
n∈Z
4.4 we see that A = k[Z, Z −1 ], so X = Gm,k .
Example 4.6. Let X = {z ∈ K | |q| ≤ |z| ≤ 1} with q ∈ K × and |q| < 1. We
have
q
A = KhZ, U i/(ZU − q) = KhZ, i
Z




X
X ³ q ´n ¯¯
¯ lim |an | = lim |bn | = 0 ,
= f=
an Z n +
bn
¯ n→∞
n→∞


Z
n≥0
n>0
and kf k = max {|an1 |, |bn2 |}. Hence A = k[Z 0 , U 0 ]/Z 0 U 0 , where Z 0 is the image of
n1 ,n2
Z and U 0 is the image of q/Z. The analytic reduction X consists of two affine lines
`1 and `2 intersecting (transversally) at one point P . The points in X with |z| = 1
are mapped onto `1 − {P }, and the points with |z| = |q| are mapped onto `2 − {P }.
Finally, the points which satisfy |q| < |z| < 1 are mapped to P . This is an analytic
incarnation of a deformation of a node.
×
with |q| < 1. For the affinoid
Example 4.7. Let X = Gan
m,K . Take q ∈ K
S
covering n∈Z Xn in Example 3.12, X is an infinite chain of P1k ’s. This follows
from the previous example. Indeed, X n consists of two affine lines `1,n and `2,n
intersecting at a point Pn . The reduction X n+1 is glued to X n by identifying
`2,n − {Pn } and `1,n+1 − {Pn+1 } so that there results P1k with P1k − {0} = `2,n and
P1k −{∞} = ­`1,n+1 . To see this last®part note that the affinoid algebra corresponding
to Xn is K q −n/2 Z, q (n+1)/2 Z −1 , and Xn is glued to Xn+1 via q (n+1)/2 Z −1 7→
q −(n+1)/2 Z. On the other hand, q (n+1)/2 Z −1 gives the parameter u on `2,n =
Spec(k[u]), whereas q −(n+1)/2 Z gives the parameter z on `1,n+1 = Spec(k[z]), cf.
Example 4.6. Hence `1,n+1 is glued to `2,n via u 7→ z = 1/u.
g
Example 4.8. The analytic reduction of T an = (Gan
m,K ) is the product sitg
uation of the previous example: T an = X with X = Gan
m,K . Each irreducible
component of T an is isomorphic to the product of g copies of P1k ’s. The irreducible
components are glued together along unions of coordinate hyperplanes in k g . In
particular, every singular point in T an is locally isomorphic to a singularity in the
intersection of d coordinate hyperplanes in k g Swith some (varying) d ≤ g. If we
remove the singular locus from T an then we get α∈Zg (Ggm,k )α , i.e., a disjoint union
of copies of Ggm,k labelled by g-tuples of integers.
Example 4.9. Let G = T an /Λ. We use the affinoid covering of T an in §4.2.
By construction, this affinoid covering is preserved under the action of Λ. Hence
the group Λ acts equivariantly on the analytic reduction T an → T an . The analytic
reduction of T an /Λ is a finite union of copies of g-tuple products (P1k ) × · · · × (P1k )
glued together along coordinate hyperplanes, and the smooth locus of G is an
extension of a finite abelian group by Ggm,k .
As a more concrete example of this, let’s consider the analytic reduction of
Z
an
the Tate curve E = Gan
m,K /q . Refine the affinoid covering {Xi }i∈Z of Gm,K given
in Example 3.12, so that q acts by Xn 7→ Xn+2m for some fixed m ∈ Z+ (this
is easily done by appropriately decreasing the radii of Xi ). Then q acts on the
infinite chain {P1k }i∈Z forming the reduction of Gan
m,K by m-shifts, i.e., by mapping
RIGID-ANALYTIC GEOMETRY
13
(P1k )i identically onto (P1k )i+m . The analytic reduction E is a union of m copies of
P1k , arranged as the sides of a polygon, and intersecting each other transversally (if
m = 1 then there is one P1k forming a node); see Figure 1.
Xn
mod q Z
Gan
m,K
E
an. red.
an. red.
P1k
mod q Z
P1k
Gan
m,K
E
Figure 1. Analytic reductions of Gan
m,K and E
4.4. Formal schemes and their generic fibres. There is a close and very
useful relationship (discovered by Raynaud) between formal schemes over R and
rigid spaces over K. We will indicate this relationship first making it explicit for
the affinoid space Sp(Tn ). Consider the polynomial ring P = R[Z1 , . . . , Zn ]. The
formal completion of P with respect to the ideal mP is, by definition, the projective
limit lim(P/mm P ). This R-algebra is denoted by Tn = RhZ1 , . . . , Zn i. It consists
←
P
of the power series s∈Nn cs Z1s1 · · · Znsn with all cs ∈ R and such that for every
m > 0 there are only finitely many s’s with cs 6∈ mm R. That is, this is the set
of elements in Tn with coefficients in R. The affine formal scheme Spf(Tn ) is the
ringed space with underlying topological space Spec k[Z1 , . . . , Zn ] and the structure
sheaf RhZ1 , . . . , Zn i; this is the “formal n-ball”. The connection to Sp(Tn ) results
from the isomorphism K ⊗R Tn = Tn . Now we extend this construction to a functor
from (certain) formal schemes over R to rigid spaces over K.
Let A be a reduced affinoid algebra. The R-algebra A◦ can be proved to be madically separated and complete, and in fact a quotient of RhZ1 , . . . , Zn i for some
n, so the formal scheme Spf(A◦ ) may be realized as a closed formal subscheme
14
MIHRAN PAPIKIAN
of a “formal n-ball” over Spf(R) for some n ≥ 0. The underlying topological
space of Spf(A◦ ) is the analytic reduction Spec(A), and Sp(K ⊗R A◦ ) = Sp(A)
is (by definition) the rigid space associated to Spf(A◦ ). This construction can be
extended to formal schemes which are separated and locally of finite type over R.
One covers the formal scheme X by formal affines Spf(A), and glues Sp(A ⊗R K)’s.
The resulting rigid-analytic space is called the generic fibre of X, and is denoted by
Xrig . One can check that X → Xrig is functorial; see [BL2].
Given a separated scheme X over R which admits a locally finite affine covering,
we now have two ways to associate a rigid-analytic space to it. First, we could
consider the generic fibre XK := X ×R K of X, which is a locally finite type Kan
. Second, we could consider the formal
scheme, and take its analytification XK
completion X of X along its closed fibre (i.e., the formal completion of X with
respect to an ideal of definition m of R), and then take its generic fibre Xrig . In
general, there is a morphism
(4.1)
an
iX : Xrig → XK
which is an open immersion, and it is an isomorphism for proper X over R (essentially because of the valuative criterion for properness); see Theorem 5.3.1 in
[Co2].
Example 4.10. Let X = AnR . In this case X ×R K = AnK , hence (XK )an =
is the analytification of the n-dimensional affine space over K. The completion
along the closed fibre is Spf(Tn ), so Xrig is the unit polydisc Sp(Tn ). It is easy to
give an explicit admissible covering of X an such that Xrig is one of the affinoids in
that covering; see Example 3.11 for the case n = 1.
An,an
K
Example 4.11. The most important example for us is when X = Gm,R , a split
torus over R, so (X ×R K)an = Gan
m,K is the analytic one-dimensional torus over
−1
K and X = Spf(RhZ, Z i). Hence Xrig = Sp(KhZ, Z −1 i). In this case iX is the
open immersion of the “unit circle” into the “punctured plane”.
Finally, we recall Grothendieck’s formal GAGA theorem from Chapter 5 of
EGA III, which we will need in §4.5. Let X be a proper scheme over Spec(R),
b denote the formal completion of X along m. Grothendieck’s theorem is
and let X
b is a fully faithful functor. Moreover, if X is proper
two-fold: first of all, X → X
over Spf(R) and L is an invertible OX -module such that L0 := L/mL is ample on
X0 = X ×Spf(R) Spec(k) then there is a proper scheme X over Spec(R) such that
b In this latter case we say X is algebraic.
X∼
= X.
4.5. Néron models. Assume K is a complete, discretely valued field. We
return to our initial goal of characterizing the abelian varieties representable as
quotients of analytic tori by lattices. First we recall the notion of Néron model,
which plays an important role in our characterization.
Definition 4.12. Let A be an abelian variety over K. A Néron model of
A is a smooth, separated, finite type scheme A over R such that AK ∼
= A and
which satisfies the following universal property, called Néron mapping property: for
each smooth R-scheme B and each K-morphism uK : BK → A there is a unique
R-morphism u : B → A extending uK .
It is clear from the Néron mapping property that if A exists then it is unique,
and has a unique R-group scheme structure which extends the K-group scheme
RIGID-ANALYTIC GEOMETRY
15
structure on A. It is a non-trivial theorem (due to Néron) that A always exists; see
[BLR].
Let G = T an /Λ and let U be the explicit affinoid covering used to give G its
analytic structure as in Example 4.9. As we discussed in §4.4, we can construct a
formal scheme G, using the covering U , such that Grig ∼
= G (recall that G is proper).
Assume G is algebraic. Using Grothendieck’s GAGA, one can deduce that G is also
algebraic. Let G be the proper scheme over Spec(R) such that Gb ∼
= G. There is an
an ∼
isomorphism GK
= G.
= G via (4.1). Let A be the abelian variety for which Aan ∼
Then G is a model of A over R. Let G 0 be the scheme over R which is obtained by
removing the singular locus of Gk from G. The R-scheme G 0 is smooth, separated,
of finite type, and the identity section (extending that on A) lies in G 0 . Its closed
fibre is an extension of a finite abelian group by Ggm,k ; c.f. Example 4.9. Let A be
the Néron model of A over R. By the Néron mapping property the isomorphism
∼
0
GK
−→ AK uniquely extends to an R-morphism G 0 → A. The image of G 0 is open
in the fibres of A. Hence the connected component of the identity A0k of Ak is
isomorphic to the split algebraic torus Ggm,k . Abelian varieties whose Néron models
have this property are called totally degenerate. We conclude that abelian varieties
which admit analytic uniformization must be totally degenerate. (Being totally
degenerate is a rather special property. For example, the abelian varieties which
extend to abelian schemes over Spec(R) do not have this property.)
The property of being “totally degenerate” is also sufficient for the existence
of an analytic uniformization in the sense discussed above. To see how this goes,
let A be a totally degenerate abelian variety of dimension g over K. We apply
(4.1) to the relative connected component of the identity A0 of A, that is, to the
largest open subscheme of A containing the identity section which has connected
fibres. Since A0K = A, on the right side of (4.1) we have the analytification Aan of
g
A. On the other hand, A0k ∼
= Gm,k . The rigidity of tori (cf. Theorem 3.6 in Exposé
IX of SGA 3) implies that the formal completion of A0 along its closed fibre is
b g = (Spf RhZ, Z −1 i)g respecting
uniquely isomorphic to the formal split torus G
m
g
0 ∼
a choice of isomorphism Ak = Gm,k . Thus, by Example 4.11 we get an open
b g )rig ,→ Aan . We also have the analytic torus
immersion of analytic groups iA0 : (G
m
an
an
g
g rig
b
b g )rig ,→ T an . The
T = (Gm,K ) associated to (Gm ) and an open immersion (G
m
key fact is that iA0 extends uniquely to a rigid-analytic group morphism T an → Aan ,
and its kernel is a lattice Λ ⊂ (K × )g of rank g. This induces an isomorphism of
analytic groups T an /Λ ∼
= Aan ; see Theorem 1.2 in [BL1].
T anD
v;
v
D
v
D
vv
v
D
v
D!
, vv
iA 0
/ Aan
b g )rig
(G
m
We record all of this into a theorem:
Theorem 4.13. An abelian variety over K admits analytic uniformization (in
the sense of §4.2) if and only if it is totally degenerate.
Note that totally degenerate abelian varieties are essentially the ones considered
in Mumford’s fundamental paper [M3], except that Mumford allowed a normal
complete base with dimension possibly greater than 1 and so he had to use formal
16
MIHRAN PAPIKIAN
geometry throughout. For a 1-dimensional base, his work can be motivated by the
above picture from rigid-analytic geometry; cf. [BL1].
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[BGR]
[BL1]
[BL2]
[BLR]
[Ca]
[CF]
[Co1]
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[D]
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[LRS]
[M1]
[M2]
[M3]
[OvdP]
[Ra]
[Ri]
[S1]
[S2]
[Si1]
[Si2]
[T1]
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QQ
Department of Mathematics, Stanford University, Stanford, CA 94305
E-mail address: papikian@math.stanford.edu